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Unformatted text preview: Impact of a sphere with a flat surface A sphere of mass m and radius R is impacting a flat rigid surface with the initial velocity v . The sphere has the elastic modulus E 1 and the rigid flat surface has the elastic modulus E 2 = E 1 = E . The equivalent elastic modulus E is calculated from 1 E = 1- 2 1 E 1 + 1- 2 2 E 2 or E = E 2(1- 2 ) , (1) where 1 and 2 are Poissons ratios of the two materials in contact 1 = 2 = (same material). The equivalent radius for two spheres in contact is 1 R = 1 R 1 + 1 R 2 . (2) For the the collision of the sphere with the radius R 1 against the rigid flat surface with the radius R 2 the equivalent radius is R = R 1 . An effective mass for two bodies in contact can be obtained from the defini- tion 1 m = 1 m 1 + 1 m 2 . (3) For the the collision of the sphere with the the mass m 1 against the rigid flat surface with the mass m 2 the effective mass is m = m 1 . The interference, x , is the distance the sphere is displaced normally into the rigid flat. The Hertz solution assumes that the interference is small enough such that the geometry does not change signifiable. The motion of the impact point during the collision can be specified by one of the following three cases: I. Elastic compression phase This case starts with the contact moment ( the contact force P is zero, P = 0) and ends when the contact force reaches the known value of the critical force ( P = P c ). For the critical force the deformation is the critical deformation ( x = x c ). During this case there are only elastic deformations x = x e ( x e x c ) and the Hertz law is applied P = k 1 x 3 / 2 , (4) 1 where k 1 = 4 3 E R = 2 E R 3(1- 2 ) . (5) The equation of motion of the sphere is m x = mg- P or m x = mg- k 1 x 3 / 2 . (6) The initial conditions are at t =0, x (0)=0, and x (0)= v...
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- Fall '11